Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until h...

This Complex Functions Theory a-4 text is the fourth e-book in a series which has previously characterized analytic functions by their complex differentiability and proved Cauchy’s Integral Theorem, provided alternative proofs which show that locally, every analytic function is described by its Taylor series, shown the connection between analytic functions and geometry, and reviewed conformal maps and their importance in solving Dirichlet problems. Complex Functions Theory a-4 builds on these previous texts, focusing on the general theory of the Laplace Transformation Operator. This e-book and previous titles in the series can be downloaded for free here.

All theorems are accompanied by their proofs, and all equations are explained and demonstrated in detail. A comprehensive index follows the text.

Readers interested in a full overview of complex analytic functions should refer to the related titles in this series, all of which are available for free download on bookboon.com: Elementary Analytic Functions - Complex Functions Theory a-1, Calculus of Residua - Complex Functions Theory a-2, Stability, Riemann Surfaces, and Conformal Mappings: Complex Functions Theory a-3.

The Lebesgue Integral

Null sets and null functions

The Lebesgue integral

The Laplace transformation

Definition of the Laplace transformation using complex functions theory

Some important properties of Laplace transforms

The complex inversion formula I

Convolutions

Linear ordinary differential equations

Other transformations and the general inversion formula

The two-sided Laplace transformation

The Fourier transformation

The Fourier transformation on L1(R)

The Mellin transformation

The complex inversion formula II

Laplace transformation of series

A catalogue of methods of finding the Laplace transform and the inverse Laplace transform